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Gupta, Neha
- Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under HÖlder Continuity Condition
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Authors
Neha Gupta
1,
J. P. Jaiswal
1
Affiliations
1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, IN
1 Department of Mathematics, Maulana Azad National Institute of Technology, Bhopal, Madhya Pradesh - 462003, IN
Source
The Journal of the Indian Mathematical Society, Vol 87, No 1-2 (2020), Pagination: 56–69Abstract
The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x* and its uniqueness along with prior error bound are established. Also, the R-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.Keywords
Banach Space, Nonlinear Operator, Semilocal Convergence, Hölder Condition, Frèchet Derivative, Recurrence Relation, Error Bound.References
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Authors
Source
The Journal of the Indian Mathematical Society, Vol 89, No 3-4 (2022), Pagination: 305-316Abstract
The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. $\omega$-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution.
Keywords
Banach Space, Lcal Cconvergence, Recurrence Relation, β-Continuity condition.References
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